The authors investigate oscillatory properties of the linear Hamiltonian system
where are -matrices with continuous entries for , are symmetric and the matrix is supposed to be positive definite. Using a transformation which preserves the oscillatory nature of transformed systems, system (*) is transformed into the “diagonal off” system , with positive definite, hence this system is equivalent to the second-order vector-matrix Sturm-Liouville differential equation
The main result of the paper is formulated under the assumption that the matrix is nonnegative definite ( denotes the identity matrix). Under this assumption, system (**) is a Sturmian majorant of the system . So, oscillation of the last system implies oscillation of (**) and hence, in turn, also of (*). From this point of view, the results of the paper are very close to those presented in the paper of G. J. Butler, L. H. Erbe and A. B. Mingarelli [Trans. Am. Math. Soc. 303, 263–282 (1987; Zbl 0648.34031)].